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Let $X$ be a hyperbolic manifold of finite volume.

I want to prove that $X$ has ends of the form $N\times \mathbb{R}$ where $N$ has a finite covering by a nilmanifold and $\pi_1N\to \pi_1 X$ is injective.

I have no idea about it. Why does $X$ have splitting ends and why does it have injectivity about fundamental groups? Could you please give some help with the detail? Thank you very much!

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    $\begingroup$ check the internet for "Margulis lemma" and "thin-thick decomposition". $\endgroup$
    – Uri Bader
    Commented Oct 19, 2021 at 6:32
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    $\begingroup$ Chapter IV of Benedetti-Petronio „Lectures on Hyperbolic Geometry“ has a complete proof. $\endgroup$
    – ThiKu
    Commented Oct 19, 2021 at 7:11
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    $\begingroup$ If "hyperbolic" means "real hyperbolic", $N$ even has a finite covering by a torus. $\endgroup$
    – YCor
    Commented Oct 19, 2021 at 8:49

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